// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (c) 2021, NVIDIA CORPORATION. All rights reserved.
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MATHFUNCTIONS_H
#define EIGEN_MATHFUNCTIONS_H

// TODO this should better be moved to NumTraits
// Source: WolframAlpha
#define EIGEN_PI 3.141592653589793238462643383279502884197169399375105820974944592307816406L
#define EIGEN_LOG2E 1.442695040888963407359924681001892137426645954152985934135449406931109219L
#define EIGEN_LN2 0.693147180559945309417232121458176568075500134360255254120680009493393621L

namespace Eigen {

// On WINCE, std::abs is defined for int only, so let's defined our own overloads:
// This issue has been confirmed with MSVC 2008 only, but the issue might exist for more recent versions too.
#if EIGEN_OS_WINCE && EIGEN_COMP_MSVC && EIGEN_COMP_MSVC <= 1500
long abs(long x) { return (labs(x)); }
double abs(double x) { return (fabs(x)); }
float abs(float x) { return (fabsf(x)); }
long double abs(long double x) { return (fabsl(x)); }
#endif

namespace internal {

    /** \internal \class global_math_functions_filtering_base
  *
  * What it does:
  * Defines a typedef 'type' as follows:
  * - if type T has a member typedef Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl, then
  *   global_math_functions_filtering_base<T>::type is a typedef for it.
  * - otherwise, global_math_functions_filtering_base<T>::type is a typedef for T.
  *
  * How it's used:
  * To allow to defined the global math functions (like sin...) in certain cases, like the Array expressions.
  * When you do sin(array1+array2), the object array1+array2 has a complicated expression type, all what you want to know
  * is that it inherits ArrayBase. So we implement a partial specialization of sin_impl for ArrayBase<Derived>.
  * So we must make sure to use sin_impl<ArrayBase<Derived> > and not sin_impl<Derived>, otherwise our partial specialization
  * won't be used. How does sin know that? That's exactly what global_math_functions_filtering_base tells it.
  *
  * How it's implemented:
  * SFINAE in the style of enable_if. Highly susceptible of breaking compilers. With GCC, it sure does work, but if you replace
  * the typename dummy by an integer template parameter, it doesn't work anymore!
  */

    template <typename T, typename dummy = void> struct global_math_functions_filtering_base
    {
        typedef T type;
    };

    template <typename T> struct always_void
    {
        typedef void type;
    };

    template <typename T>
    struct global_math_functions_filtering_base<T, typename always_void<typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl>::type>
    {
        typedef typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl type;
    };

#define EIGEN_MATHFUNC_IMPL(func, scalar) Eigen::internal::func##_impl<typename Eigen::internal::global_math_functions_filtering_base<scalar>::type>
#define EIGEN_MATHFUNC_RETVAL(func, scalar) \
    typename Eigen::internal::func##_retval<typename Eigen::internal::global_math_functions_filtering_base<scalar>::type>::type

    /****************************************************************************
* Implementation of real                                                 *
****************************************************************************/

    template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex> struct real_default_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return x; }
    };

    template <typename Scalar> struct real_default_impl<Scalar, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x)
        {
            using std::real;
            return real(x);
        }
    };

    template <typename Scalar> struct real_impl : real_default_impl<Scalar>
    {
    };

#if defined(EIGEN_GPU_COMPILE_PHASE)
    template <typename T> struct real_impl<std::complex<T>>
    {
        typedef T RealScalar;
        EIGEN_DEVICE_FUNC
        static inline T run(const std::complex<T>& x) { return x.real(); }
    };
#endif

    template <typename Scalar> struct real_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of imag                                                 *
****************************************************************************/

    template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex> struct imag_default_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar&) { return RealScalar(0); }
    };

    template <typename Scalar> struct imag_default_impl<Scalar, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x)
        {
            using std::imag;
            return imag(x);
        }
    };

    template <typename Scalar> struct imag_impl : imag_default_impl<Scalar>
    {
    };

#if defined(EIGEN_GPU_COMPILE_PHASE)
    template <typename T> struct imag_impl<std::complex<T>>
    {
        typedef T RealScalar;
        EIGEN_DEVICE_FUNC
        static inline T run(const std::complex<T>& x) { return x.imag(); }
    };
#endif

    template <typename Scalar> struct imag_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of real_ref                                             *
****************************************************************************/

    template <typename Scalar> struct real_ref_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar& run(Scalar& x) { return reinterpret_cast<RealScalar*>(&x)[0]; }
        EIGEN_DEVICE_FUNC
        static inline const RealScalar& run(const Scalar& x) { return reinterpret_cast<const RealScalar*>(&x)[0]; }
    };

    template <typename Scalar> struct real_ref_retval
    {
        typedef typename NumTraits<Scalar>::Real& type;
    };

    /****************************************************************************
* Implementation of imag_ref                                             *
****************************************************************************/

    template <typename Scalar, bool IsComplex> struct imag_ref_default_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar& run(Scalar& x) { return reinterpret_cast<RealScalar*>(&x)[1]; }
        EIGEN_DEVICE_FUNC
        static inline const RealScalar& run(const Scalar& x) { return reinterpret_cast<RealScalar*>(&x)[1]; }
    };

    template <typename Scalar> struct imag_ref_default_impl<Scalar, false>
    {
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static inline Scalar run(Scalar&) { return Scalar(0); }
        EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static inline const Scalar run(const Scalar&) { return Scalar(0); }
    };

    template <typename Scalar> struct imag_ref_impl : imag_ref_default_impl<Scalar, NumTraits<Scalar>::IsComplex>
    {
    };

    template <typename Scalar> struct imag_ref_retval
    {
        typedef typename NumTraits<Scalar>::Real& type;
    };

    /****************************************************************************
* Implementation of conj                                                 *
****************************************************************************/

    template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex> struct conj_default_impl
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x) { return x; }
    };

    template <typename Scalar> struct conj_default_impl<Scalar, true>
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x)
        {
            using std::conj;
            return conj(x);
        }
    };

    template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex> struct conj_impl : conj_default_impl<Scalar, IsComplex>
    {
    };

    template <typename Scalar> struct conj_retval
    {
        typedef Scalar type;
    };

    /****************************************************************************
* Implementation of abs2                                                 *
****************************************************************************/

    template <typename Scalar, bool IsComplex> struct abs2_impl_default
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return x * x; }
    };

    template <typename Scalar> struct abs2_impl_default<Scalar, true>  // IsComplex
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return x.real() * x.real() + x.imag() * x.imag(); }
    };

    template <typename Scalar> struct abs2_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return abs2_impl_default<Scalar, NumTraits<Scalar>::IsComplex>::run(x); }
    };

    template <typename Scalar> struct abs2_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of sqrt/rsqrt                                             *
****************************************************************************/

    template <typename Scalar> struct sqrt_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_ALWAYS_INLINE Scalar run(const Scalar& x)
        {
            EIGEN_USING_STD(sqrt);
            return sqrt(x);
        }
    };

    // Complex sqrt defined in MathFunctionsImpl.h.
    template <typename T> EIGEN_DEVICE_FUNC std::complex<T> complex_sqrt(const std::complex<T>& a_x);

    // Custom implementation is faster than `std::sqrt`, works on
    // GPU, and correctly handles special cases (unlike MSVC).
    template <typename T> struct sqrt_impl<std::complex<T>>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_ALWAYS_INLINE std::complex<T> run(const std::complex<T>& x) { return complex_sqrt<T>(x); }
    };

    template <typename Scalar> struct sqrt_retval
    {
        typedef Scalar type;
    };

    // Default implementation relies on numext::sqrt, at bottom of file.
    template <typename T> struct rsqrt_impl;

    // Complex rsqrt defined in MathFunctionsImpl.h.
    template <typename T> EIGEN_DEVICE_FUNC std::complex<T> complex_rsqrt(const std::complex<T>& a_x);

    template <typename T> struct rsqrt_impl<std::complex<T>>
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_ALWAYS_INLINE std::complex<T> run(const std::complex<T>& x) { return complex_rsqrt<T>(x); }
    };

    template <typename Scalar> struct rsqrt_retval
    {
        typedef Scalar type;
    };

    /****************************************************************************
* Implementation of norm1                                                *
****************************************************************************/

    template <typename Scalar, bool IsComplex> struct norm1_default_impl;

    template <typename Scalar> struct norm1_default_impl<Scalar, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x)
        {
            EIGEN_USING_STD(abs);
            return abs(x.real()) + abs(x.imag());
        }
    };

    template <typename Scalar> struct norm1_default_impl<Scalar, false>
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x)
        {
            EIGEN_USING_STD(abs);
            return abs(x);
        }
    };

    template <typename Scalar> struct norm1_impl : norm1_default_impl<Scalar, NumTraits<Scalar>::IsComplex>
    {
    };

    template <typename Scalar> struct norm1_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of hypot                                                *
****************************************************************************/

    template <typename Scalar> struct hypot_impl;

    template <typename Scalar> struct hypot_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of cast                                                 *
****************************************************************************/

    template <typename OldType, typename NewType, typename EnableIf = void> struct cast_impl
    {
        EIGEN_DEVICE_FUNC
        static inline NewType run(const OldType& x) { return static_cast<NewType>(x); }
    };

    // Casting from S -> Complex<T> leads to an implicit conversion from S to T,
    // generating warnings on clang.  Here we explicitly cast the real component.
    template <typename OldType, typename NewType>
    struct cast_impl<OldType, NewType, typename internal::enable_if<!NumTraits<OldType>::IsComplex && NumTraits<NewType>::IsComplex>::type>
    {
        EIGEN_DEVICE_FUNC
        static inline NewType run(const OldType& x)
        {
            typedef typename NumTraits<NewType>::Real NewReal;
            return static_cast<NewType>(static_cast<NewReal>(x));
        }
    };

    // here, for once, we're plainly returning NewType: we don't want cast to do weird things.

    template <typename OldType, typename NewType> EIGEN_DEVICE_FUNC inline NewType cast(const OldType& x) { return cast_impl<OldType, NewType>::run(x); }

    /****************************************************************************
* Implementation of round                                                   *
****************************************************************************/

    template <typename Scalar> struct round_impl
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex), NUMERIC_TYPE_MUST_BE_REAL)
#if EIGEN_HAS_CXX11_MATH
            EIGEN_USING_STD(round);
#endif
            return Scalar(round(x));
        }
    };

#if !EIGEN_HAS_CXX11_MATH
#if EIGEN_HAS_C99_MATH
    // Use ::roundf for float.
    template <> struct round_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static inline float run(const float& x) { return ::roundf(x); }
    };
#else
    template <typename Scalar> struct round_using_floor_ceil_impl
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex), NUMERIC_TYPE_MUST_BE_REAL)
            // Without C99 round/roundf, resort to floor/ceil.
            EIGEN_USING_STD(floor);
            EIGEN_USING_STD(ceil);
            // If not enough precision to resolve a decimal at all, return the input.
            // Otherwise, adding 0.5 can trigger an increment by 1.
            const Scalar limit = Scalar(1ull << (NumTraits<Scalar>::digits() - 1));
            if (x >= limit || x <= -limit)
            {
                return x;
            }
            return (x > Scalar(0)) ? Scalar(floor(x + Scalar(0.5))) : Scalar(ceil(x - Scalar(0.5)));
        }
    };

    template <> struct round_impl<float> : round_using_floor_ceil_impl<float>
    {
    };

    template <> struct round_impl<double> : round_using_floor_ceil_impl<double>
    {
    };
#endif  // EIGEN_HAS_C99_MATH
#endif  // !EIGEN_HAS_CXX11_MATH

    template <typename Scalar> struct round_retval
    {
        typedef Scalar type;
    };

    /****************************************************************************
* Implementation of rint                                                    *
****************************************************************************/

    template <typename Scalar> struct rint_impl
    {
        EIGEN_DEVICE_FUNC
        static inline Scalar run(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex), NUMERIC_TYPE_MUST_BE_REAL)
#if EIGEN_HAS_CXX11_MATH
            EIGEN_USING_STD(rint);
#endif
            return rint(x);
        }
    };

#if !EIGEN_HAS_CXX11_MATH
    template <> struct rint_impl<double>
    {
        EIGEN_DEVICE_FUNC
        static inline double run(const double& x) { return ::rint(x); }
    };
    template <> struct rint_impl<float>
    {
        EIGEN_DEVICE_FUNC
        static inline float run(const float& x) { return ::rintf(x); }
    };
#endif

    template <typename Scalar> struct rint_retval
    {
        typedef Scalar type;
    };

/****************************************************************************
* Implementation of arg                                                     *
****************************************************************************/

// Visual Studio 2017 has a bug where arg(float) returns 0 for negative inputs.
// This seems to be fixed in VS 2019.
#if EIGEN_HAS_CXX11_MATH && (!EIGEN_COMP_MSVC || EIGEN_COMP_MSVC >= 1920)
    // std::arg is only defined for types of std::complex, or integer types or float/double/long double
    template <typename Scalar,
              bool HasStdImpl = NumTraits<Scalar>::IsComplex || is_integral<Scalar>::value || is_same<Scalar, float>::value || is_same<Scalar, double>::value ||
                                is_same<Scalar, long double>::value>
    struct arg_default_impl;

    template <typename Scalar> struct arg_default_impl<Scalar, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x)
        {
#if defined(EIGEN_HIP_DEVICE_COMPILE)
            // HIP does not seem to have a native device side implementation for the math routine "arg"
            using std::arg;
#else
            EIGEN_USING_STD(arg);
#endif
            return static_cast<RealScalar>(arg(x));
        }
    };

    // Must be non-complex floating-point type (e.g. half/bfloat16).
    template <typename Scalar> struct arg_default_impl<Scalar, false>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return (x < Scalar(0)) ? RealScalar(EIGEN_PI) : RealScalar(0); }
    };
#else
    template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex> struct arg_default_impl
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x) { return (x < RealScalar(0)) ? RealScalar(EIGEN_PI) : RealScalar(0); }
    };

    template <typename Scalar> struct arg_default_impl<Scalar, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        EIGEN_DEVICE_FUNC
        static inline RealScalar run(const Scalar& x)
        {
            EIGEN_USING_STD(arg);
            return arg(x);
        }
    };
#endif
    template <typename Scalar> struct arg_impl : arg_default_impl<Scalar>
    {
    };

    template <typename Scalar> struct arg_retval
    {
        typedef typename NumTraits<Scalar>::Real type;
    };

    /****************************************************************************
* Implementation of expm1                                                   *
****************************************************************************/

    // This implementation is based on GSL Math's expm1.
    namespace std_fallback {
        // fallback expm1 implementation in case there is no expm1(Scalar) function in namespace of Scalar,
        // or that there is no suitable std::expm1 function available. Implementation
        // attributed to Kahan. See: http://www.plunk.org/~hatch/rightway.php.
        template <typename Scalar> EIGEN_DEVICE_FUNC inline Scalar expm1(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
            typedef typename NumTraits<Scalar>::Real RealScalar;

            EIGEN_USING_STD(exp);
            Scalar u = exp(x);
            if (numext::equal_strict(u, Scalar(1)))
            {
                return x;
            }
            Scalar um1 = u - RealScalar(1);
            if (numext::equal_strict(um1, Scalar(-1)))
            {
                return RealScalar(-1);
            }

            EIGEN_USING_STD(log);
            Scalar logu = log(u);
            return numext::equal_strict(u, logu) ? u : (u - RealScalar(1)) * x / logu;
        }
    }  // namespace std_fallback

    template <typename Scalar> struct expm1_impl
    {
        EIGEN_DEVICE_FUNC static inline Scalar run(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
#if EIGEN_HAS_CXX11_MATH
            using std::expm1;
#else
            using std_fallback::expm1;
#endif
            return expm1(x);
        }
    };

    template <typename Scalar> struct expm1_retval
    {
        typedef Scalar type;
    };

    /****************************************************************************
* Implementation of log                                                     *
****************************************************************************/

    // Complex log defined in MathFunctionsImpl.h.
    template <typename T> EIGEN_DEVICE_FUNC std::complex<T> complex_log(const std::complex<T>& z);

    template <typename Scalar> struct log_impl
    {
        EIGEN_DEVICE_FUNC static inline Scalar run(const Scalar& x)
        {
            EIGEN_USING_STD(log);
            return static_cast<Scalar>(log(x));
        }
    };

    template <typename Scalar> struct log_impl<std::complex<Scalar>>
    {
        EIGEN_DEVICE_FUNC static inline std::complex<Scalar> run(const std::complex<Scalar>& z) { return complex_log(z); }
    };

    /****************************************************************************
* Implementation of log1p                                                   *
****************************************************************************/

    namespace std_fallback {
        // fallback log1p implementation in case there is no log1p(Scalar) function in namespace of Scalar,
        // or that there is no suitable std::log1p function available
        template <typename Scalar> EIGEN_DEVICE_FUNC inline Scalar log1p(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
            typedef typename NumTraits<Scalar>::Real RealScalar;
            EIGEN_USING_STD(log);
            Scalar x1p = RealScalar(1) + x;
            Scalar log_1p = log_impl<Scalar>::run(x1p);
            const bool is_small = numext::equal_strict(x1p, Scalar(1));
            const bool is_inf = numext::equal_strict(x1p, log_1p);
            return (is_small || is_inf) ? x : x * (log_1p / (x1p - RealScalar(1)));
        }
    }  // namespace std_fallback

    template <typename Scalar> struct log1p_impl
    {
        EIGEN_DEVICE_FUNC static inline Scalar run(const Scalar& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
#if EIGEN_HAS_CXX11_MATH
            using std::log1p;
#else
            using std_fallback::log1p;
#endif
            return log1p(x);
        }
    };

    // Specialization for complex types that are not supported by std::log1p.
    template <typename RealScalar> struct log1p_impl<std::complex<RealScalar>>
    {
        EIGEN_DEVICE_FUNC static inline std::complex<RealScalar> run(const std::complex<RealScalar>& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(RealScalar)
            return std_fallback::log1p(x);
        }
    };

    template <typename Scalar> struct log1p_retval
    {
        typedef Scalar type;
    };

    /****************************************************************************
* Implementation of pow                                                  *
****************************************************************************/

    template <typename ScalarX, typename ScalarY, bool IsInteger = NumTraits<ScalarX>::IsInteger&& NumTraits<ScalarY>::IsInteger> struct pow_impl
    {
        //typedef Scalar retval;
        typedef typename ScalarBinaryOpTraits<ScalarX, ScalarY, internal::scalar_pow_op<ScalarX, ScalarY>>::ReturnType result_type;
        static EIGEN_DEVICE_FUNC inline result_type run(const ScalarX& x, const ScalarY& y)
        {
            EIGEN_USING_STD(pow);
            return pow(x, y);
        }
    };

    template <typename ScalarX, typename ScalarY> struct pow_impl<ScalarX, ScalarY, true>
    {
        typedef ScalarX result_type;
        static EIGEN_DEVICE_FUNC inline ScalarX run(ScalarX x, ScalarY y)
        {
            ScalarX res(1);
            eigen_assert(!NumTraits<ScalarY>::IsSigned || y >= 0);
            if (y & 1)
                res *= x;
            y >>= 1;
            while (y)
            {
                x *= x;
                if (y & 1)
                    res *= x;
                y >>= 1;
            }
            return res;
        }
    };

    /****************************************************************************
* Implementation of random                                               *
****************************************************************************/

    template <typename Scalar, bool IsComplex, bool IsInteger> struct random_default_impl
    {
    };

    template <typename Scalar> struct random_impl : random_default_impl<Scalar, NumTraits<Scalar>::IsComplex, NumTraits<Scalar>::IsInteger>
    {
    };

    template <typename Scalar> struct random_retval
    {
        typedef Scalar type;
    };

    template <typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random(const Scalar& x, const Scalar& y);
    template <typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random();

    template <typename Scalar> struct random_default_impl<Scalar, false, false>
    {
        static inline Scalar run(const Scalar& x, const Scalar& y) { return x + (y - x) * Scalar(std::rand()) / Scalar(RAND_MAX); }
        static inline Scalar run() { return run(Scalar(NumTraits<Scalar>::IsSigned ? -1 : 0), Scalar(1)); }
    };

    enum
    {
        meta_floor_log2_terminate,
        meta_floor_log2_move_up,
        meta_floor_log2_move_down,
        meta_floor_log2_bogus
    };

    template <unsigned int n, int lower, int upper> struct meta_floor_log2_selector
    {
        enum
        {
            middle = (lower + upper) / 2,
            value = (upper <= lower + 1) ?
                        int(meta_floor_log2_terminate) :
                        (n < (1 << middle)) ? int(meta_floor_log2_move_down) : (n == 0) ? int(meta_floor_log2_bogus) : int(meta_floor_log2_move_up)
        };
    };

    template <unsigned int n, int lower = 0, int upper = sizeof(unsigned int) * CHAR_BIT - 1, int selector = meta_floor_log2_selector<n, lower, upper>::value>
    struct meta_floor_log2
    {
    };

    template <unsigned int n, int lower, int upper> struct meta_floor_log2<n, lower, upper, meta_floor_log2_move_down>
    {
        enum
        {
            value = meta_floor_log2<n, lower, meta_floor_log2_selector<n, lower, upper>::middle>::value
        };
    };

    template <unsigned int n, int lower, int upper> struct meta_floor_log2<n, lower, upper, meta_floor_log2_move_up>
    {
        enum
        {
            value = meta_floor_log2<n, meta_floor_log2_selector<n, lower, upper>::middle, upper>::value
        };
    };

    template <unsigned int n, int lower, int upper> struct meta_floor_log2<n, lower, upper, meta_floor_log2_terminate>
    {
        enum
        {
            value = (n >= ((unsigned int)(1) << (lower + 1))) ? lower + 1 : lower
        };
    };

    template <unsigned int n, int lower, int upper> struct meta_floor_log2<n, lower, upper, meta_floor_log2_bogus>
    {
        // no value, error at compile time
    };

    template <typename Scalar> struct random_default_impl<Scalar, false, true>
    {
        static inline Scalar run(const Scalar& x, const Scalar& y)
        {
            if (y <= x)
                return x;
            // ScalarU is the unsigned counterpart of Scalar, possibly Scalar itself.
            typedef typename make_unsigned<Scalar>::type ScalarU;
            // ScalarX is the widest of ScalarU and unsigned int.
            // We'll deal only with ScalarX and unsigned int below thus avoiding signed
            // types and arithmetic and signed overflows (which are undefined behavior).
            typedef typename conditional<(ScalarU(-1) > unsigned(-1)), ScalarU, unsigned>::type ScalarX;
            // The following difference doesn't overflow, provided our integer types are two's
            // complement and have the same number of padding bits in signed and unsigned variants.
            // This is the case in most modern implementations of C++.
            ScalarX range = ScalarX(y) - ScalarX(x);
            ScalarX offset = 0;
            ScalarX divisor = 1;
            ScalarX multiplier = 1;
            const unsigned rand_max = RAND_MAX;
            if (range <= rand_max)
                divisor = (rand_max + 1) / (range + 1);
            else
                multiplier = 1 + range / (rand_max + 1);
            // Rejection sampling.
            do
            {
                offset = (unsigned(std::rand()) * multiplier) / divisor;
            } while (offset > range);
            return Scalar(ScalarX(x) + offset);
        }

        static inline Scalar run()
        {
#ifdef EIGEN_MAKING_DOCS
            return run(Scalar(NumTraits<Scalar>::IsSigned ? -10 : 0), Scalar(10));
#else
            enum {rand_bits = meta_floor_log2<(unsigned int)(RAND_MAX) + 1>::value,
                  scalar_bits = sizeof(Scalar) * CHAR_BIT,
                  shift = EIGEN_PLAIN_ENUM_MAX(0, int(rand_bits) - int(scalar_bits)),
                  offset = NumTraits<Scalar>::IsSigned ? (1 << (EIGEN_PLAIN_ENUM_MIN(rand_bits, scalar_bits) - 1)) : 0};
            return Scalar((std::rand() >> shift) - offset);
#endif
        }
    };

    template <typename Scalar> struct random_default_impl<Scalar, true, false>
    {
        static inline Scalar run(const Scalar& x, const Scalar& y) { return Scalar(random(x.real(), y.real()), random(x.imag(), y.imag())); }
        static inline Scalar run()
        {
            typedef typename NumTraits<Scalar>::Real RealScalar;
            return Scalar(random<RealScalar>(), random<RealScalar>());
        }
    };

    template <typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random(const Scalar& x, const Scalar& y)
    {
        return EIGEN_MATHFUNC_IMPL(random, Scalar)::run(x, y);
    }

    template <typename Scalar> inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random() { return EIGEN_MATHFUNC_IMPL(random, Scalar)::run(); }

// Implementation of is* functions

// std::is* do not work with fast-math and gcc, std::is* are available on MSVC 2013 and newer, as well as in clang.
#if (EIGEN_HAS_CXX11_MATH && !(EIGEN_COMP_GNUC_STRICT && __FINITE_MATH_ONLY__)) || (EIGEN_COMP_MSVC >= 1800) || (EIGEN_COMP_CLANG)
#define EIGEN_USE_STD_FPCLASSIFY 1
#else
#define EIGEN_USE_STD_FPCLASSIFY 0
#endif

    template <typename T> EIGEN_DEVICE_FUNC typename internal::enable_if<internal::is_integral<T>::value, bool>::type isnan_impl(const T&) { return false; }

    template <typename T> EIGEN_DEVICE_FUNC typename internal::enable_if<internal::is_integral<T>::value, bool>::type isinf_impl(const T&) { return false; }

    template <typename T> EIGEN_DEVICE_FUNC typename internal::enable_if<internal::is_integral<T>::value, bool>::type isfinite_impl(const T&) { return true; }

    template <typename T>
    EIGEN_DEVICE_FUNC typename internal::enable_if<(!internal::is_integral<T>::value) && (!NumTraits<T>::IsComplex), bool>::type isfinite_impl(const T& x)
    {
#if defined(EIGEN_GPU_COMPILE_PHASE)
        return (::isfinite)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
        using std::isfinite;
        return isfinite EIGEN_NOT_A_MACRO(x);
#else
        return x <= NumTraits<T>::highest() && x >= NumTraits<T>::lowest();
#endif
    }

    template <typename T>
    EIGEN_DEVICE_FUNC typename internal::enable_if<(!internal::is_integral<T>::value) && (!NumTraits<T>::IsComplex), bool>::type isinf_impl(const T& x)
    {
#if defined(EIGEN_GPU_COMPILE_PHASE)
        return (::isinf)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
        using std::isinf;
        return isinf EIGEN_NOT_A_MACRO(x);
#else
        return x > NumTraits<T>::highest() || x < NumTraits<T>::lowest();
#endif
    }

    template <typename T>
    EIGEN_DEVICE_FUNC typename internal::enable_if<(!internal::is_integral<T>::value) && (!NumTraits<T>::IsComplex), bool>::type isnan_impl(const T& x)
    {
#if defined(EIGEN_GPU_COMPILE_PHASE)
        return (::isnan)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
        using std::isnan;
        return isnan EIGEN_NOT_A_MACRO(x);
#else
        return x != x;
#endif
    }

#if (!EIGEN_USE_STD_FPCLASSIFY)

#if EIGEN_COMP_MSVC

    template <typename T> EIGEN_DEVICE_FUNC bool isinf_msvc_helper(T x) { return _fpclass(x) == _FPCLASS_NINF || _fpclass(x) == _FPCLASS_PINF; }

    //MSVC defines a _isnan builtin function, but for double only
    EIGEN_DEVICE_FUNC inline bool isnan_impl(const long double& x) { return _isnan(x) != 0; }
    EIGEN_DEVICE_FUNC inline bool isnan_impl(const double& x) { return _isnan(x) != 0; }
    EIGEN_DEVICE_FUNC inline bool isnan_impl(const float& x) { return _isnan(x) != 0; }

    EIGEN_DEVICE_FUNC inline bool isinf_impl(const long double& x) { return isinf_msvc_helper(x); }
    EIGEN_DEVICE_FUNC inline bool isinf_impl(const double& x) { return isinf_msvc_helper(x); }
    EIGEN_DEVICE_FUNC inline bool isinf_impl(const float& x) { return isinf_msvc_helper(x); }

#elif (defined __FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ && EIGEN_COMP_GNUC)

#if EIGEN_GNUC_AT_LEAST(5, 0)
#define EIGEN_TMP_NOOPT_ATTRIB EIGEN_DEVICE_FUNC inline __attribute__((optimize("no-finite-math-only")))
#else
// NOTE the inline qualifier and noinline attribute are both needed: the former is to avoid linking issue (duplicate symbol),
//      while the second prevent too aggressive optimizations in fast-math mode:
#define EIGEN_TMP_NOOPT_ATTRIB EIGEN_DEVICE_FUNC inline __attribute__((noinline, optimize("no-finite-math-only")))
#endif

    template <> EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const long double& x) { return __builtin_isnan(x); }
    template <> EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const double& x) { return __builtin_isnan(x); }
    template <> EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const float& x) { return __builtin_isnan(x); }
    template <> EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const double& x) { return __builtin_isinf(x); }
    template <> EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const float& x) { return __builtin_isinf(x); }
    template <> EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const long double& x) { return __builtin_isinf(x); }

#undef EIGEN_TMP_NOOPT_ATTRIB

#endif

#endif

    // The following overload are defined at the end of this file
    template <typename T> EIGEN_DEVICE_FUNC bool isfinite_impl(const std::complex<T>& x);
    template <typename T> EIGEN_DEVICE_FUNC bool isnan_impl(const std::complex<T>& x);
    template <typename T> EIGEN_DEVICE_FUNC bool isinf_impl(const std::complex<T>& x);

    template <typename T> T generic_fast_tanh_float(const T& a_x);
}  // end namespace internal

/****************************************************************************
* Generic math functions                                                    *
****************************************************************************/

namespace numext {

#if (!defined(EIGEN_GPUCC) || defined(EIGEN_CONSTEXPR_ARE_DEVICE_FUNC))
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T mini(const T& x, const T& y)
    {
        EIGEN_USING_STD(min)
        return min EIGEN_NOT_A_MACRO(x, y);
    }

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T maxi(const T& x, const T& y)
    {
        EIGEN_USING_STD(max)
        return max EIGEN_NOT_A_MACRO(x, y);
    }
#else
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T mini(const T& x, const T& y) { return y < x ? y : x; }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float mini(const float& x, const float& y) { return fminf(x, y); }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double mini(const double& x, const double& y) { return fmin(x, y); }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double mini(const long double& x, const long double& y)
    {
#if defined(EIGEN_HIPCC)
        // no "fminl" on HIP yet
        return (x < y) ? x : y;
#else
        return fminl(x, y);
#endif
    }

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T maxi(const T& x, const T& y) { return x < y ? y : x; }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float maxi(const float& x, const float& y) { return fmaxf(x, y); }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double maxi(const double& x, const double& y) { return fmax(x, y); }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double maxi(const long double& x, const long double& y)
    {
#if defined(EIGEN_HIPCC)
        // no "fmaxl" on HIP yet
        return (x > y) ? x : y;
#else
        return fmaxl(x, y);
#endif
    }
#endif

#if defined(SYCL_DEVICE_ONLY)

#define SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_char)  \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_short) \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_int)   \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_long)
#define SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_char)  \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_short) \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_int)   \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_long)
#define SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_uchar)   \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_ushort)  \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_uint)    \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_ulong)
#define SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_uchar)   \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_ushort)  \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_uint)    \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_ulong)
#define SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(NAME, FUNC)    \
    SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
    SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY(NAME, FUNC)
#define SYCL_SPECIALIZE_INTEGER_TYPES_UNARY(NAME, FUNC)    \
    SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
    SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY(NAME, FUNC)
#define SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(NAME, FUNC)       \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_float) \
    SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_double)
#define SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(NAME, FUNC)       \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_float) \
    SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_double)
#define SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(NAME, FUNC, RET_TYPE) \
    SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, cl::sycl::cl_float)     \
    SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, cl::sycl::cl_double)

#define SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE) \
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE RET_TYPE NAME(const ARG_TYPE& x) { return cl::sycl::FUNC(x); }

#define SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, TYPE) SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, TYPE, TYPE)

#define SYCL_SPECIALIZE_GEN1_BINARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE1, ARG_TYPE2) \
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE RET_TYPE NAME(const ARG_TYPE1& x, const ARG_TYPE2& y) { return cl::sycl::FUNC(x, y); }

#define SYCL_SPECIALIZE_GEN2_BINARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE) SYCL_SPECIALIZE_GEN1_BINARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE, ARG_TYPE)

#define SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, TYPE) SYCL_SPECIALIZE_GEN2_BINARY_FUNC(NAME, FUNC, TYPE, TYPE)

    SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(mini, min)
    SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(mini, fmin)
    SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(maxi, max)
    SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(maxi, fmax)

#endif

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(real, Scalar) real(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(real, Scalar)::run(x);
    }

    template <typename Scalar>
    EIGEN_DEVICE_FUNC inline typename internal::add_const_on_value_type<EIGEN_MATHFUNC_RETVAL(real_ref, Scalar)>::type real_ref(const Scalar& x)
    {
        return internal::real_ref_impl<Scalar>::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(real_ref, Scalar) real_ref(Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(real_ref, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(imag, Scalar) imag(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(imag, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(arg, Scalar) arg(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(arg, Scalar)::run(x);
    }

    template <typename Scalar>
    EIGEN_DEVICE_FUNC inline typename internal::add_const_on_value_type<EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar)>::type imag_ref(const Scalar& x)
    {
        return internal::imag_ref_impl<Scalar>::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar) imag_ref(Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(imag_ref, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(conj, Scalar) conj(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(conj, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(abs2, Scalar) abs2(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(abs2, Scalar)::run(x);
    }

    EIGEN_DEVICE_FUNC
    inline bool abs2(bool x) { return x; }

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T absdiff(const T& x, const T& y) { return x > y ? x - y : y - x; }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float absdiff(const float& x, const float& y) { return fabsf(x - y); }
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double absdiff(const double& x, const double& y) { return fabs(x - y); }

#if !defined(EIGEN_GPUCC)
    // HIP and CUDA do not support long double.
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double absdiff(const long double& x, const long double& y) { return fabsl(x - y); }
#endif

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(norm1, Scalar) norm1(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(norm1, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(hypot, Scalar) hypot(const Scalar& x, const Scalar& y)
    {
        return EIGEN_MATHFUNC_IMPL(hypot, Scalar)::run(x, y);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(hypot, hypot)
#endif

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(log1p, Scalar) log1p(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(log1p, Scalar)::run(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(log1p, log1p)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float log1p(const float& x) { return ::log1pf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double log1p(const double& x) { return ::log1p(x); }
#endif

    template <typename ScalarX, typename ScalarY>
    EIGEN_DEVICE_FUNC inline typename internal::pow_impl<ScalarX, ScalarY>::result_type pow(const ScalarX& x, const ScalarY& y)
    {
        return internal::pow_impl<ScalarX, ScalarY>::run(x, y);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(pow, pow)
#endif

    template <typename T> EIGEN_DEVICE_FUNC bool(isnan)(const T& x) { return internal::isnan_impl(x); }
    template <typename T> EIGEN_DEVICE_FUNC bool(isinf)(const T& x) { return internal::isinf_impl(x); }
    template <typename T> EIGEN_DEVICE_FUNC bool(isfinite)(const T& x) { return internal::isfinite_impl(x); }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isnan, isnan, bool)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isinf, isinf, bool)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isfinite, isfinite, bool)
#endif

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(rint, Scalar) rint(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(rint, Scalar)::run(x);
    }

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(round, Scalar) round(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(round, Scalar)::run(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(round, round)
#endif

    template <typename T> EIGEN_DEVICE_FUNC T(floor)(const T& x)
    {
        EIGEN_USING_STD(floor)
        return floor(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(floor, floor)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float floor(const float& x) { return ::floorf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double floor(const double& x) { return ::floor(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC T(ceil)(const T& x)
    {
        EIGEN_USING_STD(ceil);
        return ceil(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(ceil, ceil)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float ceil(const float& x) { return ::ceilf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double ceil(const double& x) { return ::ceil(x); }
#endif

    /** Log base 2 for 32 bits positive integers.
  * Conveniently returns 0 for x==0. */
    inline int log2(int x)
    {
        eigen_assert(x >= 0);
        unsigned int v(x);
        static const int table[32] = {0, 9, 1, 10, 13, 21, 2, 29, 11, 14, 16, 18, 22, 25, 3, 30, 8, 12, 20, 28, 15, 17, 24, 7, 19, 27, 23, 6, 26, 5, 4, 31};
        v |= v >> 1;
        v |= v >> 2;
        v |= v >> 4;
        v |= v >> 8;
        v |= v >> 16;
        return table[(v * 0x07C4ACDDU) >> 27];
    }

    /** \returns the square root of \a x.
  *
  * It is essentially equivalent to
  * \code using std::sqrt; return sqrt(x); \endcode
  * but slightly faster for float/double and some compilers (e.g., gcc), thanks to
  * specializations when SSE is enabled.
  *
  * It's usage is justified in performance critical functions, like norm/normalize.
  */
    template <typename Scalar> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE EIGEN_MATHFUNC_RETVAL(sqrt, Scalar) sqrt(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(sqrt, Scalar)::run(x);
    }

    // Boolean specialization, avoids implicit float to bool conversion (-Wimplicit-conversion-floating-point-to-bool).
    template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_DEVICE_FUNC bool sqrt<bool>(const bool& x) { return x; }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sqrt, sqrt)
#endif

    /** \returns the reciprocal square root of \a x. **/
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T rsqrt(const T& x) { return internal::rsqrt_impl<T>::run(x); }

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T log(const T& x) { return internal::log_impl<T>::run(x); }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(log, log)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float log(const float& x) { return ::logf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double log(const double& x) { return ::log(x); }
#endif

    template <typename T>
    EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE typename internal::enable_if<NumTraits<T>::IsSigned || NumTraits<T>::IsComplex, typename NumTraits<T>::Real>::type
    abs(const T& x)
    {
        EIGEN_USING_STD(abs);
        return abs(x);
    }

    template <typename T>
    EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE typename internal::enable_if<!(NumTraits<T>::IsSigned || NumTraits<T>::IsComplex), typename NumTraits<T>::Real>::type
    abs(const T& x)
    {
        return x;
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_INTEGER_TYPES_UNARY(abs, abs)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(abs, fabs)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float abs(const float& x) { return ::fabsf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double abs(const double& x) { return ::fabs(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float abs(const std::complex<float>& x) { return ::hypotf(x.real(), x.imag()); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double abs(const std::complex<double>& x) { return ::hypot(x.real(), x.imag()); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T exp(const T& x)
    {
        EIGEN_USING_STD(exp);
        return exp(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(exp, exp)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float exp(const float& x) { return ::expf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double exp(const double& x) { return ::exp(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE std::complex<float> exp(const std::complex<float>& x)
    {
        float com = ::expf(x.real());
        float res_real = com * ::cosf(x.imag());
        float res_imag = com * ::sinf(x.imag());
        return std::complex<float>(res_real, res_imag);
    }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE std::complex<double> exp(const std::complex<double>& x)
    {
        double com = ::exp(x.real());
        double res_real = com * ::cos(x.imag());
        double res_imag = com * ::sin(x.imag());
        return std::complex<double>(res_real, res_imag);
    }
#endif

    template <typename Scalar> EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(expm1, Scalar) expm1(const Scalar& x)
    {
        return EIGEN_MATHFUNC_IMPL(expm1, Scalar)::run(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(expm1, expm1)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float expm1(const float& x) { return ::expm1f(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double expm1(const double& x) { return ::expm1(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T cos(const T& x)
    {
        EIGEN_USING_STD(cos);
        return cos(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(cos, cos)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float cos(const float& x) { return ::cosf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double cos(const double& x) { return ::cos(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T sin(const T& x)
    {
        EIGEN_USING_STD(sin);
        return sin(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sin, sin)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float sin(const float& x) { return ::sinf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double sin(const double& x) { return ::sin(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T tan(const T& x)
    {
        EIGEN_USING_STD(tan);
        return tan(x);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(tan, tan)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tan(const float& x) { return ::tanf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double tan(const double& x) { return ::tan(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T acos(const T& x)
    {
        EIGEN_USING_STD(acos);
        return acos(x);
    }

#if EIGEN_HAS_CXX11_MATH
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T acosh(const T& x)
    {
        EIGEN_USING_STD(acosh);
        return static_cast<T>(acosh(x));
    }
#endif

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(acos, acos)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(acosh, acosh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float acos(const float& x) { return ::acosf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double acos(const double& x) { return ::acos(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T asin(const T& x)
    {
        EIGEN_USING_STD(asin);
        return asin(x);
    }

#if EIGEN_HAS_CXX11_MATH
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T asinh(const T& x)
    {
        EIGEN_USING_STD(asinh);
        return static_cast<T>(asinh(x));
    }
#endif

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(asin, asin)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(asinh, asinh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float asin(const float& x) { return ::asinf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double asin(const double& x) { return ::asin(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T atan(const T& x)
    {
        EIGEN_USING_STD(atan);
        return static_cast<T>(atan(x));
    }

#if EIGEN_HAS_CXX11_MATH
    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T atanh(const T& x)
    {
        EIGEN_USING_STD(atanh);
        return static_cast<T>(atanh(x));
    }
#endif

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(atan, atan)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(atanh, atanh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float atan(const float& x) { return ::atanf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double atan(const double& x) { return ::atan(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T cosh(const T& x)
    {
        EIGEN_USING_STD(cosh);
        return static_cast<T>(cosh(x));
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(cosh, cosh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float cosh(const float& x) { return ::coshf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double cosh(const double& x) { return ::cosh(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T sinh(const T& x)
    {
        EIGEN_USING_STD(sinh);
        return static_cast<T>(sinh(x));
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sinh, sinh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float sinh(const float& x) { return ::sinhf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double sinh(const double& x) { return ::sinh(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T tanh(const T& x)
    {
        EIGEN_USING_STD(tanh);
        return tanh(x);
    }

#if (!defined(EIGEN_GPUCC)) && EIGEN_FAST_MATH && !defined(SYCL_DEVICE_ONLY)
    EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tanh(float x) { return internal::generic_fast_tanh_float(x); }
#endif

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(tanh, tanh)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tanh(const float& x) { return ::tanhf(x); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double tanh(const double& x) { return ::tanh(x); }
#endif

    template <typename T> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T fmod(const T& a, const T& b)
    {
        EIGEN_USING_STD(fmod);
        return fmod(a, b);
    }

#if defined(SYCL_DEVICE_ONLY)
    SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(fmod, fmod)
#endif

#if defined(EIGEN_GPUCC)
    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float fmod(const float& a, const float& b) { return ::fmodf(a, b); }

    template <> EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double fmod(const double& a, const double& b) { return ::fmod(a, b); }
#endif

#if defined(SYCL_DEVICE_ONLY)
#undef SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_BINARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_UNARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE
#undef SYCL_SPECIALIZE_GEN_UNARY_FUNC
#undef SYCL_SPECIALIZE_UNARY_FUNC
#undef SYCL_SPECIALIZE_GEN1_BINARY_FUNC
#undef SYCL_SPECIALIZE_GEN2_BINARY_FUNC
#undef SYCL_SPECIALIZE_BINARY_FUNC
#endif

}  // end namespace numext

namespace internal {

    template <typename T> EIGEN_DEVICE_FUNC bool isfinite_impl(const std::complex<T>& x)
    {
        return (numext::isfinite)(numext::real(x)) && (numext::isfinite)(numext::imag(x));
    }

    template <typename T> EIGEN_DEVICE_FUNC bool isnan_impl(const std::complex<T>& x)
    {
        return (numext::isnan)(numext::real(x)) || (numext::isnan)(numext::imag(x));
    }

    template <typename T> EIGEN_DEVICE_FUNC bool isinf_impl(const std::complex<T>& x)
    {
        return ((numext::isinf)(numext::real(x)) || (numext::isinf)(numext::imag(x))) && (!(numext::isnan)(x));
    }

    /****************************************************************************
* Implementation of fuzzy comparisons                                       *
****************************************************************************/

    template <typename Scalar, bool IsComplex, bool IsInteger> struct scalar_fuzzy_default_impl
    {
    };

    template <typename Scalar> struct scalar_fuzzy_default_impl<Scalar, false, false>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        template <typename OtherScalar> EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const Scalar& x, const OtherScalar& y, const RealScalar& prec)
        {
            return numext::abs(x) <= numext::abs(y) * prec;
        }
        EIGEN_DEVICE_FUNC
        static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar& prec)
        {
            return numext::abs(x - y) <= numext::mini(numext::abs(x), numext::abs(y)) * prec;
        }
        EIGEN_DEVICE_FUNC
        static inline bool isApproxOrLessThan(const Scalar& x, const Scalar& y, const RealScalar& prec) { return x <= y || isApprox(x, y, prec); }
    };

    template <typename Scalar> struct scalar_fuzzy_default_impl<Scalar, false, true>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        template <typename OtherScalar> EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const Scalar& x, const Scalar&, const RealScalar&)
        {
            return x == Scalar(0);
        }
        EIGEN_DEVICE_FUNC
        static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar&) { return x == y; }
        EIGEN_DEVICE_FUNC
        static inline bool isApproxOrLessThan(const Scalar& x, const Scalar& y, const RealScalar&) { return x <= y; }
    };

    template <typename Scalar> struct scalar_fuzzy_default_impl<Scalar, true, false>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        template <typename OtherScalar> EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const Scalar& x, const OtherScalar& y, const RealScalar& prec)
        {
            return numext::abs2(x) <= numext::abs2(y) * prec * prec;
        }
        EIGEN_DEVICE_FUNC
        static inline bool isApprox(const Scalar& x, const Scalar& y, const RealScalar& prec)
        {
            return numext::abs2(x - y) <= numext::mini(numext::abs2(x), numext::abs2(y)) * prec * prec;
        }
    };

    template <typename Scalar> struct scalar_fuzzy_impl : scalar_fuzzy_default_impl<Scalar, NumTraits<Scalar>::IsComplex, NumTraits<Scalar>::IsInteger>
    {
    };

    template <typename Scalar, typename OtherScalar>
    EIGEN_DEVICE_FUNC inline bool
    isMuchSmallerThan(const Scalar& x, const OtherScalar& y, const typename NumTraits<Scalar>::Real& precision = NumTraits<Scalar>::dummy_precision())
    {
        return scalar_fuzzy_impl<Scalar>::template isMuchSmallerThan<OtherScalar>(x, y, precision);
    }

    template <typename Scalar>
    EIGEN_DEVICE_FUNC inline bool
    isApprox(const Scalar& x, const Scalar& y, const typename NumTraits<Scalar>::Real& precision = NumTraits<Scalar>::dummy_precision())
    {
        return scalar_fuzzy_impl<Scalar>::isApprox(x, y, precision);
    }

    template <typename Scalar>
    EIGEN_DEVICE_FUNC inline bool
    isApproxOrLessThan(const Scalar& x, const Scalar& y, const typename NumTraits<Scalar>::Real& precision = NumTraits<Scalar>::dummy_precision())
    {
        return scalar_fuzzy_impl<Scalar>::isApproxOrLessThan(x, y, precision);
    }

    /******************************************
***  The special case of the  bool type ***
******************************************/

    template <> struct random_impl<bool>
    {
        static inline bool run() { return random<int>(0, 1) == 0 ? false : true; }

        static inline bool run(const bool& a, const bool& b) { return random<int>(a, b) == 0 ? false : true; }
    };

    template <> struct scalar_fuzzy_impl<bool>
    {
        typedef bool RealScalar;

        template <typename OtherScalar> EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const bool& x, const bool&, const bool&) { return !x; }

        EIGEN_DEVICE_FUNC
        static inline bool isApprox(bool x, bool y, bool) { return x == y; }

        EIGEN_DEVICE_FUNC
        static inline bool isApproxOrLessThan(const bool& x, const bool& y, const bool&) { return (!x) || y; }
    };

}  // end namespace internal

// Default implementations that rely on other numext implementations
namespace internal {

    // Specialization for complex types that are not supported by std::expm1.
    template <typename RealScalar> struct expm1_impl<std::complex<RealScalar>>
    {
        EIGEN_DEVICE_FUNC static inline std::complex<RealScalar> run(const std::complex<RealScalar>& x)
        {
            EIGEN_STATIC_ASSERT_NON_INTEGER(RealScalar)
            RealScalar xr = x.real();
            RealScalar xi = x.imag();
            // expm1(z) = exp(z) - 1
            //          = exp(x +  i * y) - 1
            //          = exp(x) * (cos(y) + i * sin(y)) - 1
            //          = exp(x) * cos(y) - 1 + i * exp(x) * sin(y)
            // Imag(expm1(z)) = exp(x) * sin(y)
            // Real(expm1(z)) = exp(x) * cos(y) - 1
            //          = exp(x) * cos(y) - 1.
            //          = expm1(x) + exp(x) * (cos(y) - 1)
            //          = expm1(x) + exp(x) * (2 * sin(y / 2) ** 2)
            RealScalar erm1 = numext::expm1<RealScalar>(xr);
            RealScalar er = erm1 + RealScalar(1.);
            RealScalar sin2 = numext::sin(xi / RealScalar(2.));
            sin2 = sin2 * sin2;
            RealScalar s = numext::sin(xi);
            RealScalar real_part = erm1 - RealScalar(2.) * er * sin2;
            return std::complex<RealScalar>(real_part, er * s);
        }
    };

    template <typename T> struct rsqrt_impl
    {
        EIGEN_DEVICE_FUNC
        static EIGEN_ALWAYS_INLINE T run(const T& x) { return T(1) / numext::sqrt(x); }
    };

#if defined(EIGEN_GPU_COMPILE_PHASE)
    template <typename T> struct conj_impl<std::complex<T>, true>
    {
        EIGEN_DEVICE_FUNC
        static inline std::complex<T> run(const std::complex<T>& x) { return std::complex<T>(numext::real(x), -numext::imag(x)); }
    };
#endif

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_MATHFUNCTIONS_H
